Removing the logarithmic loss in Helmholtz Nyström estimates

Let $\Gamma \subset \mathbb{R}^2$ be a smooth embedded closed curve with a smooth periodic parametrisation $\gamma : \mathbb{R}/2\pi\mathbb{Z} \to \Gamma$ satisfying $0 < |\gamma'(t)| < c_{\max}$. For $k > 0$ consider the oscillatory kernels $$L_{1,k}(t,\tau) = k \int_{S^1} e^{ik\langle \gamma(t) - \gamma(\tau), \omega\rangle} f_k(\omega, t, \tau)\, dS(\omega),$$ where the amplitude $f_k$ is smooth, periodic in $t$ and $\tau$, and has every derivative in $(\omega, t, \tau)$ bounded independently of $k$; kernels of this form arise from the Helmholtz single- and double-layer operators in the splitting behind Kress quadrature. Let $$\widehat L_{1,k,m}(t) = \frac{1}{\sqrt{2\pi}} \int_0^{2\pi} e^{-im\tau} L_{1,k}(t,\tau)\, d\tau, \qquad \langle z \rangle = (1 + |z|^2)^{1/2},$$ regard functions of $t$ as functions on $\Gamma$, and let $H^s_k(\Gamma)$ carry the norm $\|u\|_{H^s_k} = \|(I - k^{-2}\Delta_\Gamma)^{s/2} u\|_{L^2(\Gamma)}$. Galkowski, Rachh and Spence proved the bound below when $\Gamma$ is convex with nonvanishing curvature and $\gamma$ has unit speed. For general $\Gamma$ they proved it with $C$ replaced by $C\sqrt{\log k}$, for each fixed $\epsilon$ and $N > (1+\epsilon)(1-\epsilon)^{-1} c_{\max} k$.

Problem. Is it true that for every $k_0 > 0$ and $s \in \mathbb{R}$ there is a constant $C$ such that $$\frac1k \sum_{0 < |m| \le (1-\epsilon)N} \|\widehat L_{1,k,m}\|_{H^s_k(\Gamma)} \langle m/k \rangle^s \le C$$ for all $k > k_0$, $N > 0$ and $0 < \epsilon < 1$? The constant may depend on the curve, the parametrisation, the bounds on the amplitude, $s$ and $k_0$, but not on $k$, $N$ or $\epsilon$.